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  "body": "\n## Statement\n\nLet $q \\ge 1$ and $a$ be integers, and let $y \\ge 2$ be real.\nLet $\\psi(y; q, a)$ be the sum of the [[von Mangoldt function|0NK8JTE1NN]] $\\Lambda(n)$ over $n \\le y$ with $n \\equiv a \\pmod q$.\nLet $\\varphi$ be [[Euler's totient function|0DAMHCT17R]].\nFor every $A \u003e 0$ there is $B = B(A) \u003e 0$ such that, for all $x \\ge 2$ and $Q = x^{1/2} (\\log x)^{-B}$,\n$$\\sum_{q \\le Q} \\; \\max_{\\gcd(a, q) = 1} \\; \\max_{2 \\le y \\le x} \\left| \\psi(y; q, a) - \\frac{y}{\\varphi(q)} \\right| \\le C_A \\frac{x}{(\\log x)^{A}},$$\nwhere $C_A$ depends only on $A$.\n\n## Remarks\n\nThe proof combines the [[large sieve|N15YK7XG1H]] inequality for Dirichlet characters with a decomposition of $\\Lambda$ into bilinear sums, such as Vaughan's identity; small moduli are handled by the [[Siegel–Walfisz theorem|XEPW251A97]].\nThe [[generalized Riemann hypothesis|DNKA4WBKDS]] would give $\\abs{\\psi(y; q, a) - y/\\varphi(q)} \\le C\\, y^{1/2} (\\log y)^2$ for each $q \\le y$, which yields the same bound for $Q$ up to about $x^{1/2} (\\log x)^{-A-2}$.\nSo the theorem says that GRH holds \"on average over moduli\" up to $\\sqrt{x}$.\nOne says that the primes have level of distribution $1/2$.\nThe Elliott–Halberstam conjecture states that the same bound holds with $Q = x^{\\theta}$ for every $\\theta \u003c 1$.\n\n## Explanations\n\n### L1 human\n\nPrimes are expected to split evenly among the residue classes modulo $q$ that they can reach.\nFor a single large $q$ nobody can prove a good error bound.\nThis theorem proves that, for almost all $q$ up to about $\\sqrt{x}$, the error is small, which is as good as what the generalized Riemann hypothesis would give on average.\n\n### L2 human\n\nThe Bombieri–Vinogradov theorem is the usual replacement for GRH in sieve theory.\nIt controls the remainder terms of a sieve applied to the primes, as in the proof of [[Chen's theorem|WE0RQTR68H]].\nIt is also the main input of the Goldston–Pintz–Yıldırım and Maynard methods for [[bounded gaps between primes|XMXFDWYYMD]].\nZhang's proof of bounded gaps needed a variant that goes slightly beyond the level $1/2$ for smooth moduli.\nCompare with the [[Siegel–Walfisz theorem|XEPW251A97]], which is uniform in each single modulus but only for $q$ up to a power of $\\log x$.\n",
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